So far, Eqs. (REF ) and (REF ) have been
introduced as a simple generalization of standard Brownian motion.
Importantly, however, here we do not impose any relation between the
noise strength, \(K(t)\) , and the particle's mobility, \(\mu \) . This is in
contrast to the conventional case where \(D\)  and \(\mu \)  are related via
the (2nd) FDR, that is, \(D=\mu T\) , with \(T\)  being the temperature of
the bath. For a system with time-dependent noise strength (such as
SBM with \(\alpha \ne 1\) ), the FDR is clearly broken if \(\mu \)  is
assumed to be constant. As we will proceed to show, the resulting
absence of the (conventional) FDR does not impose any problems for
several definitions and relations in standard ST
[1]}. However, complications appear when
considering the so-called medium entropy production.
